On the entropic structure of reaction-cross diffusion systems

TL;DR

This paper investigates the entropic structure of reaction-cross diffusion systems in population dynamics, establishing new existence results for weak solutions by analyzing Lyapunov functionals and introducing a novel approximation scheme.

Contribution

It provides the first comprehensive analysis of the entropic structure in reaction-cross diffusion systems with mixed convexity, along with a new approximation method for proving existence.

Findings

Existence of weak solutions for systems with convex and concave cross diffusions

Development of a new approximation scheme simplifying proofs of existence

Analysis of Lyapunov functionals for cross diffusion systems

Abstract

This paper is devoted to the study of systems of reaction-cross diffusion equations arising in population dynamics. New results of existence of weak solutions are presented, allowing to treat systems of two equations in which one of the cross diffusions is convex, while the other one is concave. The treatment of such cases involves a general study of the structure of Lyapunov functionals for cross diffusion systems, and the introduction of a new scheme of approximation, which provides simplified proofs of existence.